WE B A particle moves along a straight line. Its position, $$ x \mathrm{~m} $$, relative to a fixed point $$ O $$ on the line is given by $$ x=3+2 \sin (3 t) $$, where $$ t $$ is the time in seconds $$ (0 \leq t \leq 2 \pi) $$. a i Find the particle's greatest distance from $$ O $$. ii Find the particle's least distance from $$ O $$. b Find the times at which the particle is 5 m from $$ O $$. c Find the velocity, $$ v \mathrm{~m} / \mathrm{s} $$, and acceleration, $$ a \mathrm{~m} / \mathrm{s}^{2} $$, of the particle at time $$ t $$ seconds. d Find the maximum speed, and find the times and locations at which this occurs. e Find the maximum magnitude of the acceleration, and find the times and locations at which this occurs. f Describe the motion of the particle.

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Little Flynn · Accepted Answer

The particle's greatest distance from $$ O $$ is 5 m, and its least distance is 1 m. It is 5 m from $$ O $$ at $$ t = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{11\pi}{6} $$. The velocity is $$ v(t) = 6 \cos(3t) $$ and the acceleration is $$ a(t) = -18 \sin(3t) $$. The maximum speed is 6 m/s at $$ t = 0, \frac{2\pi}{3}, \frac{4\pi}{3}, 2\pi $$, and the maximum acceleration is 18 m/s² at $$ t = \frac{\pi}{6}, \frac{\pi}{2}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6} $$. The motion is simple harmonic motion with a period of $$ \frac{2\pi}{3} $$ seconds.

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